Question

If $f(x) = \left\{ \begin{gathered}
  mx - 1,x \leqslant 5 \\
  3x - 5,x > 5 \\
\end{gathered} \right.$ is continuous then the value of m is :
A.$\dfrac{{11}}{5}$
B.$\dfrac{5}{{11}}$
C.$\dfrac{5}{3}$
D.$\dfrac{3}{5}$

Answer 1

Hint: We are given that the function f(x) is continuous at 5 and we know that any function is continuous at x = a then $\mathop {\lim }\limits_{x \to {a^ - }} f(x) = \mathop {\lim }\limits_{x \to {a^ + }} f(x)$ and applying this we can get the value of m

Complete step-by-step answer:
The given function is $f(x) = \left\{ \begin{gathered}
  mx - 1,x \leqslant 5 \\
  3x - 5,x > 5 \\
\end{gathered} \right.$
And we are given that f(x) is continuous at x = 5
Whenever a function is continuous at x = a then
$\mathop {\lim }\limits_{x \to {a^ - }} f(x) = \mathop {\lim }\limits_{x \to {a^ + }} f(x)$
Same way, applying this in our given function
Since the value of f(x) when x is less than 5 is mx – 1
We get $\mathop {\lim }\limits_{x \to {5^ - }} f(x) = m(5) - 1 = 5m - 1$ ……….(1)
Since the value of f(x) when x is more than 5 is 3x - 5
We get $\mathop {\lim }\limits_{x \to {5^ + }} f(x) = 3(5) - 5 = 15 - 5 = 10$ ……….(2)
Equating (1) and (2)
$
   \Rightarrow 5m - 1 = 10 \\
   \Rightarrow 5m = 10 + 1 \\
   \Rightarrow m = \dfrac{{11}}{5} \\
$
The correct option is a.

Note: A function is said to be continuous if a small change in the input only causes a small change in the output.
A function is continuous when its graph is a single unbroken curve